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Improved method of multicriteria fuzzy decision-making based on vague sets
Computer-Aided Design 39 (2007) 164–169 www.elsevier.com/locate/cad
Improved method of multicriteria fuzzy decision-making based on vague sets Jun Ye ∗ Department of Mechatronic Engineering, Shaoxing College of Arts and Sciences, 508 Huancheng West Road, Shaoxing, Zhejiang Province 312000, PR China Received 1 March 2006; accepted 23 November 2006
Abstract An improved method is presented, which provides improved score functions to measure the degree of suitability of each of a set of alternatives, with respect to a set of criteria presented with vague values. The improved algorithm for score functions is introduced by taking into account the effect of an unknown degree (hesitancy degree) of the vague values on the degree of suitability to which each alternative satisfies the decisionmaker’s requirement. The meaning of the proposed function is more transparent than that of other existing functions, which are not reasonable in some cases. The proposed function illustrates that it has stronger discrimination in comparison with previous functions. The applicability of this improved multicriteria fuzzy decision-making approach is also demonstrated by means of examples. The improved method can be used to rank the decision alternatives according to the decision criteria. The functions proposed in this paper can provide a more useful technique than previous functions, in order to efficiently help the decision-maker. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Fuzzy set; Vague set; Multicriteria fuzzy decision-making
1. Introduction On many occasions, an engineering decision needs to be made based on available data and information that are vague, imprecise, and uncertain, by nature. The decision-making process of engineering schemes in the conceptual design phase is one of these typical occasions, which frequently calls upon some method of treating uncertain data and information. In a conceptual design phase, designers usually present many alternatives. However, the subjective characteristics of the alternatives are generally uncertain and need to be evaluated based on the decision-maker’s insufficient knowledge and judgments. The nature of this vagueness and uncertainty is fuzzy, rather than random, especially when subjective assessments are involved in the decision-making process. Fuzzy set theory offers a possibility for handing these sorts of data and information involving the subjective characteristics of human nature in the decision-making process. Since the theory of fuzzy sets [1] was proposed in 1965, it has been used for handling fuzzy decision-making problems ∗ Tel.: + 86 575 8327323.
E-mail address: [email protected]. c 2006 Elsevier Ltd. All rights reserved. 0010-4485/$ - see front matter doi:10.1016/j.cad.2006.11.005
[2–8,10,11]. Kickert [3] has discussed the field of fuzzy multicriteria decision-making. Zimmermann [5] illustrated a fuzzy set approach to multi-objective decision-making, and he has compared some approaches to solve multi-attribute decision-making problems based on fuzzy set theory. Yager [6] presented a fuzzy multi-attribute decision-making method that uses crisp weights, and he [7] introduced an ordered weighted aggregation operator and investigated its properties. Laarhoven et al. [8] presented a method for multi-attribute decisionmaking using fuzzy numbers as weights. Vague sets, which Cau and Buehrer [9] presented, are a generalized form of fuzzy sets. These sets were used by Chen and Tan [10]. They presented some new techniques for handling multicriteria fuzzy decision-making problems based on vague set theory, where the characteristics of the alternatives are represented by vague sets. The proposed techniques used a score function, S, to evaluate the degree of suitability to which an alternative satisfies the decision-maker’s requirement. Recently, Hong and Choi [11] proposed an accuracy function, H , to measure the degree of accuracy in the grades of membership of each alternative, with respect to a set of criteria represented by vague values. However, in some cases, these functions do not give sufficient information about alternatives.
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J. Ye / Computer-Aided Design 39 (2007) 164–169
In this paper, an improved score function is proposed to efficiently evaluate the degree of suitability of each alternative, with respect to a set of criteria represented by vague values. The proposed function can provide another useful way to help the decision-maker. The rest of this paper is organized as follows. In Section 2, we briefly describe vague set theory and the main results of Chen and Tan [10] and Hong and Choi [11]. In Section 3, we propose an improved score function and some measures to handle multicriteria fuzzy decisionmaking problems. We conclude with some final remarks of the improved method in Section 4. 2. Vague set theory and its decision-making methods 2.1. Vague sets The vague set, which is a generalization of the concept of a fuzzy set, has been introduced by Gau and Buehrer [9]. Vague set theory is introduced as follows. A vague set, A(x), in X , X = {x1 , x2 , . . . , xn }, is characterized by a truth-membership function, t A , and a falsemembership function, f A , for the elements xi ∈ X to A(x) ∈ X , (i = 1, 2, . . . , n), t A : X → [0, 1],
f A : X → [0, 1],
where the functions t A (xi ) and f A (xi ) are constrained by the condition 0 ≤ t A (xi ) + f A (xi ) ≤ 1. The truth-membership function, t A (xi ), is a lower bound on the grade of membership on the evidence for xi , and f A (xi ) is a lower bound on the negation of xi derived from the evidence against xi . The grade of membership of xi in the vague set, A, is bounded to a subinterval [t A (xi ), 1 − f A (xi )] of [0, 1]. The vague value [t A (xi ), 1 − f A (xi )] indicates that the exact grade of membership µ A (xi ) of xi may be unknown, but it is bounded by t A (xi ) ≤ µ A (xi ) ≤ 1 − f A (xi ). Fig. 1 shows a vague set in the universe of discourse, X . Let X be the universe of discourse, X = {x1 , x2 , . . . , xn }, xi ∈ X . A vague set, A, of X can be represented by A=
n X
[t A (xi ), 1 − f A (xi )]/xi ,
xi ∈ X.
i=1
For example, let X = {6, 7, 8, 9, 10}. A vague set “LARGE” of X may be defined by LARGE = [0.1, 0.2]/6 + [0.3, 0.5]/7 + [0.6, 0.8]/8 + [0.9, 1]/9 + [1, 1]/10. Note that we will omit the argument xi of t A (xi ) and f A (xi ) throughout this paper, unless they are needed for clarity. Definition 1. The intersection of two vague sets, A(x) and B(x), is a vague set, C(x), written as C(x) = A(x) ∧ B(x). The truth-membership and false-membership functions are tC and f C , respectively, where tC = min(t A , t B ), and 1 − f C = min(1 − f A , 1 − f B ). That is, [tC , 1 − f C ] = [t A , 1 − f A ] ∧ [t B , 1 − f B ] = [min(t A , t B ), min(1 − f A , 1 − f B )].
Fig. 1. A vague set.
Definition 2. The union of two vague sets, A(x) and B(x), is a vague set, C(x), written as C(x) = A(x) ∨ B(x). The truthmembership function and false-membership functions are tC and f C , respectively, where tC = max(t A , t B ), and 1 − f C = max(1 − f A , 1 − f B ). That is, [tC , 1 − f C ] = [t A , 1 − f A ] ∨ [t B , 1 − f B ] = [max(t A , t B ), max(1 − f A , 1 − f B )]. 2.2. Existing decision-making methods This section reviews the proposed functions and some measures of Chen and Tan [10], and Hong and Choi [11], to handle multi-criteria fuzzy decision-making problems. Let A be a set of alternatives and let C be a set of criteria, where A = {A1 , A2 , . . . , Am },
C = {C1 , C2 , . . . , Cn }.
Assume that the characteristics of the alternative Ai are presented by the vague set: Ai = {(C1, [ti1 , 1 − f i1 ]), (C2, [ti2 , 1 − f i2 ]), . . . , (Cn, [tin , 1 − f in ])}, where ti j indicates the degree to which the alternative Ai satisfies criteria C j , f i j indicates the degree to which the alternative Ai does not satisfy criteria C j , ti j ∈ [0, 1], f i j ∈ [0, 1], ti j + f i j ≤ 1, 1 ≤ j ≤ n, and 1 ≤ i ≤ m. Let ti∗j = 1 − f i j , where 1 ≤ j ≤ n and 1 ≤ i ≤ m. In this case, Ai can be rewritten as ∗ ∗ ∗ Ai = {(C1, [ti1 , ti1 ]), (C2, [ti2 , ti2 ]), . . . , (Cn, [tin , tin ])},
where 1 ≤ i ≤ m. In this case, the characteristics of these alternatives can be represented as shown in Table 1. Assume that there is a decision-maker who wants to choose an alternative that satisfies the criteria C j , Ck , . . . , and C p , or which satisfies the criteria Cs . This decision-maker’s requirements are represented by the following expression: C j AND Ck AND · · · AND C p OR Cs . In this case, the degrees to which the Ai satisfies or does not satisfy the decision-maker’s requirements can be measured by
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J. Ye / Computer-Aided Design 39 (2007) 164–169
Table 1 Characteristics of the alternatives
A1 A2 . . . Ai . . . Am
C1
...
Cj
...
Ck
...
Cp
...
Cs
...
Cn
∗ ] [t11 , t11 ∗ ] [t21 , t21 .. . ∗] [ti1 , ti1 .. . ∗ ] [tm1 , tm1
··· ···
[t1 j , t1∗ j ] [t2 j , t2∗ j ] .. . [ti j , ti∗j ] .. . ∗ ] [tm j , tm j
··· ···
∗ ] [t1k , t1k ∗ ] [t2k , t2k . .. ∗] [tik , tik . .. ∗ ] [tmk , tmk
··· ···
[t1 p , t1∗p ] [t2 p , t2∗p ] . .. [ti p , ti∗p ] . .. ∗ ] [tmp , tmp
··· ···
∗] [t1s , t1s ∗] [t2s , t2s . .. ∗] [tis , tis . .. ∗ ] [tms , tms
··· ···
∗ ] [t1n , t1n ∗ ] [t2n , t2n . .. ∗] [tin , tin . .. ∗ ] [tmn , tmn
··· ··· ··· ···
··· ... ··· ···
the evaluation function E,
··· ···
··· ... ··· ···
··· ... ··· ···
the decision-maker’s requirements can be represented by the following expression:
∗ ] ∧ · · · ∧ [ti p , ti∗p ]) ∨ [tis , tis∗ ] E(Ai ) = ([ti j , ti∗j ] ∧ [tik , tik
C j AND Ck AND · · · AND C p OR Cs .
∗ , . . . , ti∗p )] = [min(ti j , tik , . . . , ti p ), min(ti∗j , tik
∨[tis , tis∗ ] = [max(min(ti j , tik , . . . , ti p ), tis ), ∗ max(min(ti∗j , tik , . . . , ti∗p ), tis∗ )]
= [t Ai , t A∗ i ] = [t Ai , 1 − f Ai ],
··· ...
(1)
Assume that the weights of the criteria C j , Ck , . . . , and C p , presented by the decision-maker, are w j , wk , . . . , and w p , respectively, where w j ∈ [0, 1], wk ∈ [0, 1], . . . , w p ∈ [0, 1], and w j + wk + · · · + w p = 1. Then, the degree of suitability to which the alternative Ai satisfies the decision-maker’s requirements can be measured by the weighting function W ,
where ∧ and ∨ denote the minimum operator and the maximum operator of the vague values, respectively; E(Ai ) is a vague value, 1 ≤ i ≤ m; t Ai and t A∗ i are as follows:
∗ W (Ai ) = max{S([ti j , ti∗j ]) ∗ w j + S([tik , tik ]) ∗ wk
t Ai = max(min(ti j , tik , . . . , ti p ), tis ),
By applying Eq. (3), Eq. (4) can be rewritten as
t A∗ i
=
∗ max(min(ti∗j , tik , . . . , ti∗p ), tis∗ ).
(4)
∗ W (Ai ) = max{(ti j + ti∗j − 1) ∗ w j + (tik + tik − 1) ∗ wk
Chen and Tan [10] presented a score function S to measure the degree of suitability of each alternative, with respect to a set of criteria presented by vague values. Let x = [tx , 1 − f x ] be a vague value, where tx ∈ [0, 1], f x ∈ [0, 1], tx + f x ≤ 1. The score of x can be evaluated by the score function, S, which is shown as follows: S(x) = tx − f x ,
+ · · · + S([ti p , ti∗p ]) ∗ w p , S([tis , tis∗ ])}.
(2)
+ · · · + (ti p + ti∗p − 1) ∗ w p , (tis + tis∗ − 1)},
(5)
where W (Ai ) ∈ [−1, +1] and 1 ≤ i ≤ m. Let W (A1 ) = p1 , W (A2 ) = p2 , .. . W (Am ) = pm .
where S(x) ∈ [−1, +1]. Based on the score function, S, the degree of suitability to which the alternative Ai satisfies the decision-maker’s requirements can be measured as follows:
If W (Ai ) = pi , and pi is the largest value among the values p1 , p2 , . . . , and pm , then the alternative Ai is the best choice. We now consider the following example.
S(E(Ai )) = t Ai + t A∗ i − 1,
Example 2.1. Let A1 and A2 be two alternatives, and let C1 , C2 , and C3 be three criteria. Assuming that the characteristics of the alternatives are represented by the vague sets
(3)
where S(E(Ai )) ∈ [−1, +1]. The larger the value of S(E(Ai )), the more the suitability to which the alternative Ai (1 ≤ i ≤ m) satisfies the decision-maker’s requirement. Let S(E(A1 )) = p1 ,
A1 = {(C1 , [0, 1]), (C2 , [0, 1]), (C3 , [0, 1])}, A2 = {(C1 , [0.5, 0.5]), (C2 , [0.5, 0.5]), (C3 , [0.5, 0.5])},
S(E(A2 )) = p2 , .. .
and assuming that the decision-maker wants to choose an alternative that satisfies the criteria C1 and C2 , or which satisfies the criteria C3 , then by applying Eq. (1), we can get
S(E(Am )) = pm .
E(A1 ) = ([0, 1] ∧ [0, 1] ∨ [0, 1]) = [0, 1], E(A2 ) = ([0.5, 0.5] ∧ [0.5, 0.5] ∨ [0.5, 0.5]) = [0.5, 0.5].
If S(E(Ai )) = pi , and pi is the largest value among the values p1 , p2 , . . . , and pm , then the alternative Ai is the best choice. Assuming that the characteristics of the alternatives are shown in Table 1, and assuming that there is a decisionmaker who wants to choose an alternative which satisfies the criteria C j , Ck , . . . , and C p , or which satisfies the criteria Cs ,
By applying Eq. (3), we can get S(E(A1 )) = 0 + 1 − 1 = 0, S(E(A2 )) = 0.5 + 0.5 − 1 = 0. We do not know which is better.
J. Ye / Computer-Aided Design 39 (2007) 164–169
In Example 2.1, if the weights of the criteria C1 and C2 entered by the decision-maker are 0.7 and 0.3, respectively, then by applying Eq. (5), we can get W (A1 ) = max((0 + 1 − 1) ∗ 0.7 + (0 + 1 − 1) ∗ 0.3, 0 + 1 − 1) = max(0, 0) = 0,
In the above case, Hong and Choi [11] presented an accuracy function, H , to evaluate the degree of accuracy in the grades of membership of each alternative, with respect to a set of criteria represented by vague values. Let x = [tx , 1 − f x ] be a vague value, where tx ∈ [0, 1], f x ∈ [0, 1], tx + f x ≤ 1. The degree of accuracy of x can be evaluated by the accuracy function, H , as follows: (6)
where H (x) ∈ [0, 1]. The larger the value of H (x), the greater the degree of accuracy of the grade of membership of the vague value. Based on the accuracy function, the degree of accuracy in the grades of membership of the alternative Ai , which satisfies the decision-maker’s requirements, can be measured as follows: (7)
where H (E(Ai )) ∈ [0, 1]. The larger the value of H (E(Ai )), the greater the degree of accuracy in the grades of membership of the alternative Ai . In Example 2.1, we have obtained E(A1 ) = [0, 1] and E(A2 ) = [0.5, 0.5], with S(E(A1 )) = S(E(A2 )) = 0. Then, by applying Eq. (7), we get H (E(A1 )) = 0 + 1 − 1 = 0, H (E(A2 )) = 0.5 + 1 − 0.5 = 1. Therefore, the alternative A2 has a greater degree of accuracy than the alternative A1 in terms of grades of membership. Alternative A2 is thus the better choice. Example 2.2. Let A1 and A2 be two alternatives, and let C1 , C2 , and C3 be three criteria. Assume that the characteristics of the alternatives are represented by the vague sets A1 = {(C1 , [0.2, 0.7]), (C2 , [0.2, 0.7]), (C3 , [0.2, 0.7])}, A2 = {(C1 , [0.3, 0.8]), (C2 , [0.3, 0.8]), (C3 , [0.3, 0.8])}, and assume that the decision-maker wants to choose an alternative that satisfies the criteria C1 and C2 , or which satisfies the criterion C3 ; then by applying Eq. (1), we obtain E(A1 ) = ([02, 0.7] ∧ [0.2, 0.7] ∨ [0.2, 0.7]) = [0.2, 0.7], E(A2 ) = ([0.3, 0.8] ∧ [0.3, 0.8] ∨ [0.3, 0.8]) = [0.3, 0.8].
H (E(A1 )) = 0.2 + 1 − 0.7 = 0.5, H (E(A2 )) = 0.3 + 1 − 0.8 = 0.5.
3. Improved decision-making method
It is impossible to know which is better. What is the difference between alternative A1 and alternative A2 ? Here, some other measures are needed.
H (E(Ai )) = t Ai + 1 − t A∗ i ,
By applying Eq. (7), we obtain
In this case, we still do not know which alternative is better.
W (A2 ) = max((0.5 + 0.5 − 1) ∗ 0.7 + (0.5 + 0.5 − 1) ∗ 0.3, 0.5 + 0.5 − 1) = max(0, 0) = 0.
H (x) = tx + f x ,
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From the examples in Section 2, we can see that these functions are not reasonable in some cases. This situation can be problematic in a practical application. To overcome this problem, we propose an improved score function based on the unknown degrees of the vague values. Definition 3. Let x be a vague value, x = [tx , 1 − f x ], where tx ∈ [0, 1], f x ∈ [0, 1], tx + f x ≤ 1. The unknown degree, or hesitancy degree, of x is denoted by m x , and is defined by m x = 1 − tx − f x , and 0 ≤ m x ≤ 1. For example, let A be a vague set with truth-membership function, t A , and false-membership function, f A . If [t A , 1 − f A ] = [0.5, 0.7], then we can see that t A = 0.5, f A = 0.3, and m A = 0.2. This result can be interpreted as “the vote for resolution is 5 in favor, 3 against, and 2 abstentions”. In the following, we propose an improved score function. Let x = [tx , 1 − f x ] be a vague value, where tx ∈ [0, 1], f x ∈ [0, 1], m x ∈ [0, 1], and tx + f x ≤ 1. The score of x can be evaluated by the modified score function, J , as follows: J (x) = tx − f x + µm x = tx (1 − µ) + tx∗ (1 + µ) − 1,
(8)
where J (x) ∈ [−1, +1], and µ ∈ [−1, +1]. The parameter µ is introduced by taking into account the effect of the unknown degree m x on the score of x, and can be chosen according to actual cases. In Eq. (8), if µ > 0, the value of the third term µm x of J (x) is positive, and it tends to increase the score of x due to its addition in the first term tx ; if µ < 0, the value of the third term µm x is negative, then it tends to decrease the score of x due to its addition to the second item f x ; if µ = 0, then the improved score function J (x) is the same score function as S(x) proposed by Chen et al. [10]. In the renewed vote for resolution, µm x can also be interpreted as: most abstentions may be in favor of µ > 0, most of them for µ < 0, or they still keep abstentions with µ = 0. Based on Eqs. (1) and (8), the degree of suitability to which the alternative Ai satisfies the decision-maker’s requirements can be measured as follows: J (E(Ai )) = t Ai (1 − µi ) + t A∗ i (1 + µi ) − 1,
(9)
where J (E(Ai )) ∈ [−1, +1] and µi ∈ [−1, +1]. The larger the value of J (E(Ai )), the higher the degree of suitability to which the alternative Ai satisfies the decision-maker’s requirements. In Example 2.1, we have obtained E(A1 ) = [0, 1] and E(A2 ) = [0.5, 0.5], with S(E(A1 )) = S(E(A2 )) = 0. If µ1 and µ2 are 0.5 and 0.5, respectively, then by applying Eq. (9), we get J (E(A1 )) = 0(1 − 0.5) + 1(1 + 0.5) − 1 = 0.5, J (E(A2 )) = 0.5(1 − 0.5) + 0.5(1 + 0.5) − 1 = 0.
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J. Ye / Computer-Aided Design 39 (2007) 164–169
Therefore, the alternative A1 is better than the alternative A2 . In Example 2.2, we have obtained E(A1 ) = [0.2, 0.7] and E(A2 ) = [0.3, 0.8], with H (E(A1 )) = H (E(A2 )) = 0.5. Similarly, choosing µ1 = µ2 = 0.5 and using Eq. (9), we can get J (E(A1 )) = 0.2(1 − 0.5) + 0.7(1 + 0.5) − 1 = 0.15, J (E(A2 )) = 0.3(1 − 0.5) + 0.8(1 + 0.5) − 1 = 0.35. Obviously, the alternative A2 has a greater degree of suitability than the alternative A1 . Therefore, the alternative A2 is the better choice. This measurement method provides additional useful information to efficiently help the decision-maker. The following five theorems can be introduced according to the proposed measure function. Theorem 1. Let a = [a1 , a2 ] be a vague value, and J be the score function. Then J (a) = 0 if and only if a1 (1−µa )+a2 (1+ µa ) = 1, where µa ∈ [−1, +1]. Theorem 2. Let a = [a1 , a2 ] be a vague value, and J be the score function. Then J (a) = 1 if and only if a1 (1−µa )+a2 (1+ µa ) = 2, where µa ∈ [−1, +1]. Theorem 3. Let a = [a1 , a2 ] and b = [b1 , b2 ] be two vague values, and J be the score function. Then J (a) ≥ J (b) if and only if a1 (1 − µa ) + b2 (1 + µb ) ≥ b1 (1 − µb ) + a2 (1 + µa ), where µa ∈ [−1, +1] and µb ∈ [−1, +1]. Theorem 4. Let a = [a1 , a2 ], b = [b1 , b2 ], and c = [c1 , c2 ] be three vague values, and J be the score function. Then J (a ∧ c) ≥ J (b ∧ c) if and only if min(a1 (1 − µa ), c1 (1 − µc )) + min(b2 (1 + µb ), c2 (1 + µc )) ≥ min(b1 (1 − µb ), c1 (1 − µc )) + min(a2 (1 + µa ), c2 (1 + µc )), where µa ∈ [−1, +1], µb ∈ [−1, +1], and µc ∈ [−1, +1]. Theorem 5. Let a = [a1 , a2 ], b = [b1 , b2 ] and c = [c1 , c2 ] be three vague values, and J be the score function. Then J (a ∨ c) ≥ J (b ∨ c) if and only if max(a1 (1 − µa ), c1 (1 − µc )) + max(b2 (1 + µb ), c2 (1 + µc )) ≥ max(b1 (1 − µb ), c1 (1 − µc )) + max(a2 (1 + µa ), c2 (1 + µc )), where µa ∈ [−1, +1], µb ∈ [−1, +1], and µc ∈ [−1, +1]. The above theorems are easy to verify. The proofs are similar to those of the theorems in [10] (omitted). In the following, we present a weighted technique for handling multi-criteria fuzzy decision-making problems. Assuming that there is a decision-maker who wants to choose an alternative that satisfies the criteria C = {C1 , C2 , . . . , Ck }, or which satisfies the criteria D = {D1 , D2 , . . . , Dt }, the decision-maker’s requirements can be represented by the following expression: {C1 AND C2 AND · · · AND Ck } OR {D1 AND D2 AND · · · AND Dt } Assume that the weights of the criteria C = {C p | p = 1, 2, . . . , k} and D = {Dq |q = 1, 2, . . . , t}, entered by the decision-maker, are W = {w p | p = 1, 2, . . . , k} and
V = {vq |q = 1, 2, . . . , t}, respectively, where w p ∈ [0, 1], Pk Pt q=1 vq = 1; and p=1 w p = 1, vq ∈ [0, 1], and assume that the decision-maker enters the parameters µC p ( p = 1, 2, . . . , k) and µ Dq (q = 1, 2, . . . , t), where µC p ∈ [−1, +1] and µ Dq ∈ [−1, +1]; then the degree to which the alternativeAi satisfies the decision-maker’s requirements can be measured by the weighting function R, ∗ ∗ ]) ∗ w1 + J ([tiC2 , tiC ]) ∗ w2 R(Ai ) = max{J ([tiC1 , tiC 1 2 ∗ + · · · + J ([tiCk , tiC ]) ∗ wk , k
J ([ti D1 , ti∗D1 ]) ∗ v1 + J ([ti D2 , ti∗D2 ]) ∗ v2 + · · · + J ([ti Dt , ti∗Dt ]) ∗ vt }.
(10)
By applying Eq. (9), Eq. (10) can be rewritten into ∗ R(Ai ) = max{(tiC1 (1 − µiC1 ) + tiC (1 + µiC1 ) − 1) ∗ w1 1 ∗ (1 + µiC2 ) − 1) ∗ w2 + (tiC2 (1 − µiC2 ) + tiC 2 ∗ (1 + µiCk ) − 1) ∗ wk , + · · · + (tiCk (1 − µiCk ) + tiC k
(ti D1 (1 − µi D1 ) + ti∗D1 (1 + µi D1 ) − 1) ∗ v1 + (ti D2 (1 − µi D2 ) + ti∗D2 (1 + µi D2 ) − 1) ∗ v2 + · · · + (ti Dt (1 − µi Dt ) + ti∗Dt (1 + µi Dt ) − 1) ∗ vt }, (11) where R(Ai ) ∈ [−1, +1] and 1 ≤ i ≤ m. Let R(A1 ) = p1 , R(A2 ) = p2 , .. . R(Am ) = pm . If R(Ai ) = pi and pi is the largest value among the values p1 , p2 , . . . , and pm , then the alternative Ai is the best choice. The following example is shown in [10,11]. Example 3.1. Let A1 , A2 , A3 , A4 , and A5 be alternatives, and let C1 , C2 , and D1 be three criteria. Assuming that the characteristics of the alternatives are represented by the vague sets: A1 = {(C1 , [0.5, 0.7]), (C2 , [0.8, 0.9]), (D1 , [0.3, 0.4])}, A2 = {(C1 , [1, 1]), (C2 , [0.7, 0.8]), (D1 , [0.1, 0.2])}, A3 = {(C1 , [0, 0]), (C2 , [0.4, 0.5]), (D1 , [0.8, 0.9])}, A4 = {(C1 , [0.8, 0.9]), (C2 , [0.1, 0.2]), (D1 , [0.5, 0.6])}, A5 = {(C1 , [0.7, 0.8]), (C2 , [0.5, 0.6]), (D1 , [0.1, 0.2])}. Assume that the decision-maker wants to choose an alternative that satisfies the criteria C1 AND C2 , or which satisfies the criterion D1 , where the weights of the criteria C1 and C2 entered by the decision-maker are 0.7 and 0.3, respectively, and µiC1 , µiC2 , and µi D1 are 0.5, 0.5 and 0.5 (i = 1, 2, 3, 4, 5), respectively. By applying Eq. (8), we get J [0.5, 0.7] = 0.3000, J [0.3, 0.4] = −0.2500.
J [0.8, 0.9] = 0.7500,
J. Ye / Computer-Aided Design 39 (2007) 164–169
By applying Eqs. (10) and (11), we obtain R(A1 ) = max(J [0.5, 0.7] ∗ 0.7 + J [0.8, 0.9] ∗ 0.3, J [0.3, 0.4]) = 0.4350.
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Acknowledgment This work was supported by Natural Science Foundation of Zhejiang Province, PR China under Grant No. M603070.
By the same calculation, we get R(A2 ) = 0.8650,
R(A3 ) = 0.7500,
R(A4 ) = 0.3300,
R(A5 ) = 0.4300.
Obviously, the order of quality is R(A2 ), R(A3 ), R(A1 ), R(A5 ), and R(A4 ). Therefore, we can see that the alternative A2 is the best choice. This result is the same as in [10,11]. 4. Conclusions and further research In this paper, we have provided improved score functions to measure the degree of suitability of each alternative, with respect to a set of criteria to be represented by vague values for handling multi-criteria fuzzy decision-making problems. We also used examples to illustrate the application of the proposed function. The improved method can be used to rank the decision alternatives according to the decision criteria, and then the techniques proposed in this paper can provide more useful approaches than those of Cheng and Tan [10], and Hong and Choi [11], to efficiently help the decision-maker. But in the improved method, a new problem is how to choose a reasonable parameter µ for the improved score function. This issue will be investigated in future work.
References [1] Zadeh LA. Fuzzy sets. Information and Control 1965;8:338–56. [2] Chen SM. A new approach to handling fuzzy decision-making problems. IEEE Transactions on Systems Man and Cybernetics 1988;18:1012–6. [3] Kickert WJM. Fuzzy theories on decision making: A critical review. Boston: Kluwer Academic Publishers; 1978. [4] Jae M, Moon JH. Use of a fuzzy decision-making method in evaluating severe accident management strategies. Annals of Nuclear Energy 2002; 29:1597–606. [5] Zimmermann HJ. Fuzzy sets, decision making, and expert system. Boston: Kluwer Academic Publishers; 1987. [6] Yager RR. Fuzzy decision making including unequal objectives. Fuzzy Sets and Systems 1978;1:87–95. [7] Yager RR. On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Transactions on Systems Man and Cybernetics 1988;18:183–90. [8] Laarhoven PJM, Pedrycz W. A fuzzy extension of Saaty’s priority theory. Fuzzy Sets and Systems 1983;11:229–41. [9] Gau WL, Buehrer DJ. Vague sets. IEEE Transactions on Systems Man and Cybernetics 1993;23:610–4. [10] Chen SM, Tan JM. Handing multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets and Systems 1994;67:163–72. [11] Hong DH, Choi CH. Multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets and Systems 2000;114:103–13.
Improved method of multicriteria fuzzy decision-making based on vague sets Jun Ye ∗ Department of Mechatronic Engineering, Shaoxing College of Arts and Sciences, 508 Huancheng West Road, Shaoxing, Zhejiang Province 312000, PR China Received 1 March 2006; accepted 23 November 2006
Abstract An improved method is presented, which provides improved score functions to measure the degree of suitability of each of a set of alternatives, with respect to a set of criteria presented with vague values. The improved algorithm for score functions is introduced by taking into account the effect of an unknown degree (hesitancy degree) of the vague values on the degree of suitability to which each alternative satisfies the decisionmaker’s requirement. The meaning of the proposed function is more transparent than that of other existing functions, which are not reasonable in some cases. The proposed function illustrates that it has stronger discrimination in comparison with previous functions. The applicability of this improved multicriteria fuzzy decision-making approach is also demonstrated by means of examples. The improved method can be used to rank the decision alternatives according to the decision criteria. The functions proposed in this paper can provide a more useful technique than previous functions, in order to efficiently help the decision-maker. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Fuzzy set; Vague set; Multicriteria fuzzy decision-making
1. Introduction On many occasions, an engineering decision needs to be made based on available data and information that are vague, imprecise, and uncertain, by nature. The decision-making process of engineering schemes in the conceptual design phase is one of these typical occasions, which frequently calls upon some method of treating uncertain data and information. In a conceptual design phase, designers usually present many alternatives. However, the subjective characteristics of the alternatives are generally uncertain and need to be evaluated based on the decision-maker’s insufficient knowledge and judgments. The nature of this vagueness and uncertainty is fuzzy, rather than random, especially when subjective assessments are involved in the decision-making process. Fuzzy set theory offers a possibility for handing these sorts of data and information involving the subjective characteristics of human nature in the decision-making process. Since the theory of fuzzy sets [1] was proposed in 1965, it has been used for handling fuzzy decision-making problems ∗ Tel.: + 86 575 8327323.
E-mail address: [email protected]. c 2006 Elsevier Ltd. All rights reserved. 0010-4485/$ - see front matter doi:10.1016/j.cad.2006.11.005
[2–8,10,11]. Kickert [3] has discussed the field of fuzzy multicriteria decision-making. Zimmermann [5] illustrated a fuzzy set approach to multi-objective decision-making, and he has compared some approaches to solve multi-attribute decision-making problems based on fuzzy set theory. Yager [6] presented a fuzzy multi-attribute decision-making method that uses crisp weights, and he [7] introduced an ordered weighted aggregation operator and investigated its properties. Laarhoven et al. [8] presented a method for multi-attribute decisionmaking using fuzzy numbers as weights. Vague sets, which Cau and Buehrer [9] presented, are a generalized form of fuzzy sets. These sets were used by Chen and Tan [10]. They presented some new techniques for handling multicriteria fuzzy decision-making problems based on vague set theory, where the characteristics of the alternatives are represented by vague sets. The proposed techniques used a score function, S, to evaluate the degree of suitability to which an alternative satisfies the decision-maker’s requirement. Recently, Hong and Choi [11] proposed an accuracy function, H , to measure the degree of accuracy in the grades of membership of each alternative, with respect to a set of criteria represented by vague values. However, in some cases, these functions do not give sufficient information about alternatives.
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In this paper, an improved score function is proposed to efficiently evaluate the degree of suitability of each alternative, with respect to a set of criteria represented by vague values. The proposed function can provide another useful way to help the decision-maker. The rest of this paper is organized as follows. In Section 2, we briefly describe vague set theory and the main results of Chen and Tan [10] and Hong and Choi [11]. In Section 3, we propose an improved score function and some measures to handle multicriteria fuzzy decisionmaking problems. We conclude with some final remarks of the improved method in Section 4. 2. Vague set theory and its decision-making methods 2.1. Vague sets The vague set, which is a generalization of the concept of a fuzzy set, has been introduced by Gau and Buehrer [9]. Vague set theory is introduced as follows. A vague set, A(x), in X , X = {x1 , x2 , . . . , xn }, is characterized by a truth-membership function, t A , and a falsemembership function, f A , for the elements xi ∈ X to A(x) ∈ X , (i = 1, 2, . . . , n), t A : X → [0, 1],
f A : X → [0, 1],
where the functions t A (xi ) and f A (xi ) are constrained by the condition 0 ≤ t A (xi ) + f A (xi ) ≤ 1. The truth-membership function, t A (xi ), is a lower bound on the grade of membership on the evidence for xi , and f A (xi ) is a lower bound on the negation of xi derived from the evidence against xi . The grade of membership of xi in the vague set, A, is bounded to a subinterval [t A (xi ), 1 − f A (xi )] of [0, 1]. The vague value [t A (xi ), 1 − f A (xi )] indicates that the exact grade of membership µ A (xi ) of xi may be unknown, but it is bounded by t A (xi ) ≤ µ A (xi ) ≤ 1 − f A (xi ). Fig. 1 shows a vague set in the universe of discourse, X . Let X be the universe of discourse, X = {x1 , x2 , . . . , xn }, xi ∈ X . A vague set, A, of X can be represented by A=
n X
[t A (xi ), 1 − f A (xi )]/xi ,
xi ∈ X.
i=1
For example, let X = {6, 7, 8, 9, 10}. A vague set “LARGE” of X may be defined by LARGE = [0.1, 0.2]/6 + [0.3, 0.5]/7 + [0.6, 0.8]/8 + [0.9, 1]/9 + [1, 1]/10. Note that we will omit the argument xi of t A (xi ) and f A (xi ) throughout this paper, unless they are needed for clarity. Definition 1. The intersection of two vague sets, A(x) and B(x), is a vague set, C(x), written as C(x) = A(x) ∧ B(x). The truth-membership and false-membership functions are tC and f C , respectively, where tC = min(t A , t B ), and 1 − f C = min(1 − f A , 1 − f B ). That is, [tC , 1 − f C ] = [t A , 1 − f A ] ∧ [t B , 1 − f B ] = [min(t A , t B ), min(1 − f A , 1 − f B )].
Fig. 1. A vague set.
Definition 2. The union of two vague sets, A(x) and B(x), is a vague set, C(x), written as C(x) = A(x) ∨ B(x). The truthmembership function and false-membership functions are tC and f C , respectively, where tC = max(t A , t B ), and 1 − f C = max(1 − f A , 1 − f B ). That is, [tC , 1 − f C ] = [t A , 1 − f A ] ∨ [t B , 1 − f B ] = [max(t A , t B ), max(1 − f A , 1 − f B )]. 2.2. Existing decision-making methods This section reviews the proposed functions and some measures of Chen and Tan [10], and Hong and Choi [11], to handle multi-criteria fuzzy decision-making problems. Let A be a set of alternatives and let C be a set of criteria, where A = {A1 , A2 , . . . , Am },
C = {C1 , C2 , . . . , Cn }.
Assume that the characteristics of the alternative Ai are presented by the vague set: Ai = {(C1, [ti1 , 1 − f i1 ]), (C2, [ti2 , 1 − f i2 ]), . . . , (Cn, [tin , 1 − f in ])}, where ti j indicates the degree to which the alternative Ai satisfies criteria C j , f i j indicates the degree to which the alternative Ai does not satisfy criteria C j , ti j ∈ [0, 1], f i j ∈ [0, 1], ti j + f i j ≤ 1, 1 ≤ j ≤ n, and 1 ≤ i ≤ m. Let ti∗j = 1 − f i j , where 1 ≤ j ≤ n and 1 ≤ i ≤ m. In this case, Ai can be rewritten as ∗ ∗ ∗ Ai = {(C1, [ti1 , ti1 ]), (C2, [ti2 , ti2 ]), . . . , (Cn, [tin , tin ])},
where 1 ≤ i ≤ m. In this case, the characteristics of these alternatives can be represented as shown in Table 1. Assume that there is a decision-maker who wants to choose an alternative that satisfies the criteria C j , Ck , . . . , and C p , or which satisfies the criteria Cs . This decision-maker’s requirements are represented by the following expression: C j AND Ck AND · · · AND C p OR Cs . In this case, the degrees to which the Ai satisfies or does not satisfy the decision-maker’s requirements can be measured by
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Table 1 Characteristics of the alternatives
A1 A2 . . . Ai . . . Am
C1
...
Cj
...
Ck
...
Cp
...
Cs
...
Cn
∗ ] [t11 , t11 ∗ ] [t21 , t21 .. . ∗] [ti1 , ti1 .. . ∗ ] [tm1 , tm1
··· ···
[t1 j , t1∗ j ] [t2 j , t2∗ j ] .. . [ti j , ti∗j ] .. . ∗ ] [tm j , tm j
··· ···
∗ ] [t1k , t1k ∗ ] [t2k , t2k . .. ∗] [tik , tik . .. ∗ ] [tmk , tmk
··· ···
[t1 p , t1∗p ] [t2 p , t2∗p ] . .. [ti p , ti∗p ] . .. ∗ ] [tmp , tmp
··· ···
∗] [t1s , t1s ∗] [t2s , t2s . .. ∗] [tis , tis . .. ∗ ] [tms , tms
··· ···
∗ ] [t1n , t1n ∗ ] [t2n , t2n . .. ∗] [tin , tin . .. ∗ ] [tmn , tmn
··· ··· ··· ···
··· ... ··· ···
the evaluation function E,
··· ···
··· ... ··· ···
··· ... ··· ···
the decision-maker’s requirements can be represented by the following expression:
∗ ] ∧ · · · ∧ [ti p , ti∗p ]) ∨ [tis , tis∗ ] E(Ai ) = ([ti j , ti∗j ] ∧ [tik , tik
C j AND Ck AND · · · AND C p OR Cs .
∗ , . . . , ti∗p )] = [min(ti j , tik , . . . , ti p ), min(ti∗j , tik
∨[tis , tis∗ ] = [max(min(ti j , tik , . . . , ti p ), tis ), ∗ max(min(ti∗j , tik , . . . , ti∗p ), tis∗ )]
= [t Ai , t A∗ i ] = [t Ai , 1 − f Ai ],
··· ...
(1)
Assume that the weights of the criteria C j , Ck , . . . , and C p , presented by the decision-maker, are w j , wk , . . . , and w p , respectively, where w j ∈ [0, 1], wk ∈ [0, 1], . . . , w p ∈ [0, 1], and w j + wk + · · · + w p = 1. Then, the degree of suitability to which the alternative Ai satisfies the decision-maker’s requirements can be measured by the weighting function W ,
where ∧ and ∨ denote the minimum operator and the maximum operator of the vague values, respectively; E(Ai ) is a vague value, 1 ≤ i ≤ m; t Ai and t A∗ i are as follows:
∗ W (Ai ) = max{S([ti j , ti∗j ]) ∗ w j + S([tik , tik ]) ∗ wk
t Ai = max(min(ti j , tik , . . . , ti p ), tis ),
By applying Eq. (3), Eq. (4) can be rewritten as
t A∗ i
=
∗ max(min(ti∗j , tik , . . . , ti∗p ), tis∗ ).
(4)
∗ W (Ai ) = max{(ti j + ti∗j − 1) ∗ w j + (tik + tik − 1) ∗ wk
Chen and Tan [10] presented a score function S to measure the degree of suitability of each alternative, with respect to a set of criteria presented by vague values. Let x = [tx , 1 − f x ] be a vague value, where tx ∈ [0, 1], f x ∈ [0, 1], tx + f x ≤ 1. The score of x can be evaluated by the score function, S, which is shown as follows: S(x) = tx − f x ,
+ · · · + S([ti p , ti∗p ]) ∗ w p , S([tis , tis∗ ])}.
(2)
+ · · · + (ti p + ti∗p − 1) ∗ w p , (tis + tis∗ − 1)},
(5)
where W (Ai ) ∈ [−1, +1] and 1 ≤ i ≤ m. Let W (A1 ) = p1 , W (A2 ) = p2 , .. . W (Am ) = pm .
where S(x) ∈ [−1, +1]. Based on the score function, S, the degree of suitability to which the alternative Ai satisfies the decision-maker’s requirements can be measured as follows:
If W (Ai ) = pi , and pi is the largest value among the values p1 , p2 , . . . , and pm , then the alternative Ai is the best choice. We now consider the following example.
S(E(Ai )) = t Ai + t A∗ i − 1,
Example 2.1. Let A1 and A2 be two alternatives, and let C1 , C2 , and C3 be three criteria. Assuming that the characteristics of the alternatives are represented by the vague sets
(3)
where S(E(Ai )) ∈ [−1, +1]. The larger the value of S(E(Ai )), the more the suitability to which the alternative Ai (1 ≤ i ≤ m) satisfies the decision-maker’s requirement. Let S(E(A1 )) = p1 ,
A1 = {(C1 , [0, 1]), (C2 , [0, 1]), (C3 , [0, 1])}, A2 = {(C1 , [0.5, 0.5]), (C2 , [0.5, 0.5]), (C3 , [0.5, 0.5])},
S(E(A2 )) = p2 , .. .
and assuming that the decision-maker wants to choose an alternative that satisfies the criteria C1 and C2 , or which satisfies the criteria C3 , then by applying Eq. (1), we can get
S(E(Am )) = pm .
E(A1 ) = ([0, 1] ∧ [0, 1] ∨ [0, 1]) = [0, 1], E(A2 ) = ([0.5, 0.5] ∧ [0.5, 0.5] ∨ [0.5, 0.5]) = [0.5, 0.5].
If S(E(Ai )) = pi , and pi is the largest value among the values p1 , p2 , . . . , and pm , then the alternative Ai is the best choice. Assuming that the characteristics of the alternatives are shown in Table 1, and assuming that there is a decisionmaker who wants to choose an alternative which satisfies the criteria C j , Ck , . . . , and C p , or which satisfies the criteria Cs ,
By applying Eq. (3), we can get S(E(A1 )) = 0 + 1 − 1 = 0, S(E(A2 )) = 0.5 + 0.5 − 1 = 0. We do not know which is better.
J. Ye / Computer-Aided Design 39 (2007) 164–169
In Example 2.1, if the weights of the criteria C1 and C2 entered by the decision-maker are 0.7 and 0.3, respectively, then by applying Eq. (5), we can get W (A1 ) = max((0 + 1 − 1) ∗ 0.7 + (0 + 1 − 1) ∗ 0.3, 0 + 1 − 1) = max(0, 0) = 0,
In the above case, Hong and Choi [11] presented an accuracy function, H , to evaluate the degree of accuracy in the grades of membership of each alternative, with respect to a set of criteria represented by vague values. Let x = [tx , 1 − f x ] be a vague value, where tx ∈ [0, 1], f x ∈ [0, 1], tx + f x ≤ 1. The degree of accuracy of x can be evaluated by the accuracy function, H , as follows: (6)
where H (x) ∈ [0, 1]. The larger the value of H (x), the greater the degree of accuracy of the grade of membership of the vague value. Based on the accuracy function, the degree of accuracy in the grades of membership of the alternative Ai , which satisfies the decision-maker’s requirements, can be measured as follows: (7)
where H (E(Ai )) ∈ [0, 1]. The larger the value of H (E(Ai )), the greater the degree of accuracy in the grades of membership of the alternative Ai . In Example 2.1, we have obtained E(A1 ) = [0, 1] and E(A2 ) = [0.5, 0.5], with S(E(A1 )) = S(E(A2 )) = 0. Then, by applying Eq. (7), we get H (E(A1 )) = 0 + 1 − 1 = 0, H (E(A2 )) = 0.5 + 1 − 0.5 = 1. Therefore, the alternative A2 has a greater degree of accuracy than the alternative A1 in terms of grades of membership. Alternative A2 is thus the better choice. Example 2.2. Let A1 and A2 be two alternatives, and let C1 , C2 , and C3 be three criteria. Assume that the characteristics of the alternatives are represented by the vague sets A1 = {(C1 , [0.2, 0.7]), (C2 , [0.2, 0.7]), (C3 , [0.2, 0.7])}, A2 = {(C1 , [0.3, 0.8]), (C2 , [0.3, 0.8]), (C3 , [0.3, 0.8])}, and assume that the decision-maker wants to choose an alternative that satisfies the criteria C1 and C2 , or which satisfies the criterion C3 ; then by applying Eq. (1), we obtain E(A1 ) = ([02, 0.7] ∧ [0.2, 0.7] ∨ [0.2, 0.7]) = [0.2, 0.7], E(A2 ) = ([0.3, 0.8] ∧ [0.3, 0.8] ∨ [0.3, 0.8]) = [0.3, 0.8].
H (E(A1 )) = 0.2 + 1 − 0.7 = 0.5, H (E(A2 )) = 0.3 + 1 − 0.8 = 0.5.
3. Improved decision-making method
It is impossible to know which is better. What is the difference between alternative A1 and alternative A2 ? Here, some other measures are needed.
H (E(Ai )) = t Ai + 1 − t A∗ i ,
By applying Eq. (7), we obtain
In this case, we still do not know which alternative is better.
W (A2 ) = max((0.5 + 0.5 − 1) ∗ 0.7 + (0.5 + 0.5 − 1) ∗ 0.3, 0.5 + 0.5 − 1) = max(0, 0) = 0.
H (x) = tx + f x ,
167
From the examples in Section 2, we can see that these functions are not reasonable in some cases. This situation can be problematic in a practical application. To overcome this problem, we propose an improved score function based on the unknown degrees of the vague values. Definition 3. Let x be a vague value, x = [tx , 1 − f x ], where tx ∈ [0, 1], f x ∈ [0, 1], tx + f x ≤ 1. The unknown degree, or hesitancy degree, of x is denoted by m x , and is defined by m x = 1 − tx − f x , and 0 ≤ m x ≤ 1. For example, let A be a vague set with truth-membership function, t A , and false-membership function, f A . If [t A , 1 − f A ] = [0.5, 0.7], then we can see that t A = 0.5, f A = 0.3, and m A = 0.2. This result can be interpreted as “the vote for resolution is 5 in favor, 3 against, and 2 abstentions”. In the following, we propose an improved score function. Let x = [tx , 1 − f x ] be a vague value, where tx ∈ [0, 1], f x ∈ [0, 1], m x ∈ [0, 1], and tx + f x ≤ 1. The score of x can be evaluated by the modified score function, J , as follows: J (x) = tx − f x + µm x = tx (1 − µ) + tx∗ (1 + µ) − 1,
(8)
where J (x) ∈ [−1, +1], and µ ∈ [−1, +1]. The parameter µ is introduced by taking into account the effect of the unknown degree m x on the score of x, and can be chosen according to actual cases. In Eq. (8), if µ > 0, the value of the third term µm x of J (x) is positive, and it tends to increase the score of x due to its addition in the first term tx ; if µ < 0, the value of the third term µm x is negative, then it tends to decrease the score of x due to its addition to the second item f x ; if µ = 0, then the improved score function J (x) is the same score function as S(x) proposed by Chen et al. [10]. In the renewed vote for resolution, µm x can also be interpreted as: most abstentions may be in favor of µ > 0, most of them for µ < 0, or they still keep abstentions with µ = 0. Based on Eqs. (1) and (8), the degree of suitability to which the alternative Ai satisfies the decision-maker’s requirements can be measured as follows: J (E(Ai )) = t Ai (1 − µi ) + t A∗ i (1 + µi ) − 1,
(9)
where J (E(Ai )) ∈ [−1, +1] and µi ∈ [−1, +1]. The larger the value of J (E(Ai )), the higher the degree of suitability to which the alternative Ai satisfies the decision-maker’s requirements. In Example 2.1, we have obtained E(A1 ) = [0, 1] and E(A2 ) = [0.5, 0.5], with S(E(A1 )) = S(E(A2 )) = 0. If µ1 and µ2 are 0.5 and 0.5, respectively, then by applying Eq. (9), we get J (E(A1 )) = 0(1 − 0.5) + 1(1 + 0.5) − 1 = 0.5, J (E(A2 )) = 0.5(1 − 0.5) + 0.5(1 + 0.5) − 1 = 0.
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Therefore, the alternative A1 is better than the alternative A2 . In Example 2.2, we have obtained E(A1 ) = [0.2, 0.7] and E(A2 ) = [0.3, 0.8], with H (E(A1 )) = H (E(A2 )) = 0.5. Similarly, choosing µ1 = µ2 = 0.5 and using Eq. (9), we can get J (E(A1 )) = 0.2(1 − 0.5) + 0.7(1 + 0.5) − 1 = 0.15, J (E(A2 )) = 0.3(1 − 0.5) + 0.8(1 + 0.5) − 1 = 0.35. Obviously, the alternative A2 has a greater degree of suitability than the alternative A1 . Therefore, the alternative A2 is the better choice. This measurement method provides additional useful information to efficiently help the decision-maker. The following five theorems can be introduced according to the proposed measure function. Theorem 1. Let a = [a1 , a2 ] be a vague value, and J be the score function. Then J (a) = 0 if and only if a1 (1−µa )+a2 (1+ µa ) = 1, where µa ∈ [−1, +1]. Theorem 2. Let a = [a1 , a2 ] be a vague value, and J be the score function. Then J (a) = 1 if and only if a1 (1−µa )+a2 (1+ µa ) = 2, where µa ∈ [−1, +1]. Theorem 3. Let a = [a1 , a2 ] and b = [b1 , b2 ] be two vague values, and J be the score function. Then J (a) ≥ J (b) if and only if a1 (1 − µa ) + b2 (1 + µb ) ≥ b1 (1 − µb ) + a2 (1 + µa ), where µa ∈ [−1, +1] and µb ∈ [−1, +1]. Theorem 4. Let a = [a1 , a2 ], b = [b1 , b2 ], and c = [c1 , c2 ] be three vague values, and J be the score function. Then J (a ∧ c) ≥ J (b ∧ c) if and only if min(a1 (1 − µa ), c1 (1 − µc )) + min(b2 (1 + µb ), c2 (1 + µc )) ≥ min(b1 (1 − µb ), c1 (1 − µc )) + min(a2 (1 + µa ), c2 (1 + µc )), where µa ∈ [−1, +1], µb ∈ [−1, +1], and µc ∈ [−1, +1]. Theorem 5. Let a = [a1 , a2 ], b = [b1 , b2 ] and c = [c1 , c2 ] be three vague values, and J be the score function. Then J (a ∨ c) ≥ J (b ∨ c) if and only if max(a1 (1 − µa ), c1 (1 − µc )) + max(b2 (1 + µb ), c2 (1 + µc )) ≥ max(b1 (1 − µb ), c1 (1 − µc )) + max(a2 (1 + µa ), c2 (1 + µc )), where µa ∈ [−1, +1], µb ∈ [−1, +1], and µc ∈ [−1, +1]. The above theorems are easy to verify. The proofs are similar to those of the theorems in [10] (omitted). In the following, we present a weighted technique for handling multi-criteria fuzzy decision-making problems. Assuming that there is a decision-maker who wants to choose an alternative that satisfies the criteria C = {C1 , C2 , . . . , Ck }, or which satisfies the criteria D = {D1 , D2 , . . . , Dt }, the decision-maker’s requirements can be represented by the following expression: {C1 AND C2 AND · · · AND Ck } OR {D1 AND D2 AND · · · AND Dt } Assume that the weights of the criteria C = {C p | p = 1, 2, . . . , k} and D = {Dq |q = 1, 2, . . . , t}, entered by the decision-maker, are W = {w p | p = 1, 2, . . . , k} and
V = {vq |q = 1, 2, . . . , t}, respectively, where w p ∈ [0, 1], Pk Pt q=1 vq = 1; and p=1 w p = 1, vq ∈ [0, 1], and assume that the decision-maker enters the parameters µC p ( p = 1, 2, . . . , k) and µ Dq (q = 1, 2, . . . , t), where µC p ∈ [−1, +1] and µ Dq ∈ [−1, +1]; then the degree to which the alternativeAi satisfies the decision-maker’s requirements can be measured by the weighting function R, ∗ ∗ ]) ∗ w1 + J ([tiC2 , tiC ]) ∗ w2 R(Ai ) = max{J ([tiC1 , tiC 1 2 ∗ + · · · + J ([tiCk , tiC ]) ∗ wk , k
J ([ti D1 , ti∗D1 ]) ∗ v1 + J ([ti D2 , ti∗D2 ]) ∗ v2 + · · · + J ([ti Dt , ti∗Dt ]) ∗ vt }.
(10)
By applying Eq. (9), Eq. (10) can be rewritten into ∗ R(Ai ) = max{(tiC1 (1 − µiC1 ) + tiC (1 + µiC1 ) − 1) ∗ w1 1 ∗ (1 + µiC2 ) − 1) ∗ w2 + (tiC2 (1 − µiC2 ) + tiC 2 ∗ (1 + µiCk ) − 1) ∗ wk , + · · · + (tiCk (1 − µiCk ) + tiC k
(ti D1 (1 − µi D1 ) + ti∗D1 (1 + µi D1 ) − 1) ∗ v1 + (ti D2 (1 − µi D2 ) + ti∗D2 (1 + µi D2 ) − 1) ∗ v2 + · · · + (ti Dt (1 − µi Dt ) + ti∗Dt (1 + µi Dt ) − 1) ∗ vt }, (11) where R(Ai ) ∈ [−1, +1] and 1 ≤ i ≤ m. Let R(A1 ) = p1 , R(A2 ) = p2 , .. . R(Am ) = pm . If R(Ai ) = pi and pi is the largest value among the values p1 , p2 , . . . , and pm , then the alternative Ai is the best choice. The following example is shown in [10,11]. Example 3.1. Let A1 , A2 , A3 , A4 , and A5 be alternatives, and let C1 , C2 , and D1 be three criteria. Assuming that the characteristics of the alternatives are represented by the vague sets: A1 = {(C1 , [0.5, 0.7]), (C2 , [0.8, 0.9]), (D1 , [0.3, 0.4])}, A2 = {(C1 , [1, 1]), (C2 , [0.7, 0.8]), (D1 , [0.1, 0.2])}, A3 = {(C1 , [0, 0]), (C2 , [0.4, 0.5]), (D1 , [0.8, 0.9])}, A4 = {(C1 , [0.8, 0.9]), (C2 , [0.1, 0.2]), (D1 , [0.5, 0.6])}, A5 = {(C1 , [0.7, 0.8]), (C2 , [0.5, 0.6]), (D1 , [0.1, 0.2])}. Assume that the decision-maker wants to choose an alternative that satisfies the criteria C1 AND C2 , or which satisfies the criterion D1 , where the weights of the criteria C1 and C2 entered by the decision-maker are 0.7 and 0.3, respectively, and µiC1 , µiC2 , and µi D1 are 0.5, 0.5 and 0.5 (i = 1, 2, 3, 4, 5), respectively. By applying Eq. (8), we get J [0.5, 0.7] = 0.3000, J [0.3, 0.4] = −0.2500.
J [0.8, 0.9] = 0.7500,
J. Ye / Computer-Aided Design 39 (2007) 164–169
By applying Eqs. (10) and (11), we obtain R(A1 ) = max(J [0.5, 0.7] ∗ 0.7 + J [0.8, 0.9] ∗ 0.3, J [0.3, 0.4]) = 0.4350.
169
Acknowledgment This work was supported by Natural Science Foundation of Zhejiang Province, PR China under Grant No. M603070.
By the same calculation, we get R(A2 ) = 0.8650,
R(A3 ) = 0.7500,
R(A4 ) = 0.3300,
R(A5 ) = 0.4300.
Obviously, the order of quality is R(A2 ), R(A3 ), R(A1 ), R(A5 ), and R(A4 ). Therefore, we can see that the alternative A2 is the best choice. This result is the same as in [10,11]. 4. Conclusions and further research In this paper, we have provided improved score functions to measure the degree of suitability of each alternative, with respect to a set of criteria to be represented by vague values for handling multi-criteria fuzzy decision-making problems. We also used examples to illustrate the application of the proposed function. The improved method can be used to rank the decision alternatives according to the decision criteria, and then the techniques proposed in this paper can provide more useful approaches than those of Cheng and Tan [10], and Hong and Choi [11], to efficiently help the decision-maker. But in the improved method, a new problem is how to choose a reasonable parameter µ for the improved score function. This issue will be investigated in future work.
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